Question

A playground merry-go-round has a radius R and a rotational inertia I. When the merry-go-round is at rest, a child with mass m runs with speed v along a line tangent to the rim and jumps on. The angular velocity of the merry-go-round is then

Answer 1

Initial angular momentum of the child + merry-go-round system with respect to the center of the merry-go-round is,
Li = m * v * R

Once the child is at the rim of the merry-go-round, the moment of inertia changes,
Final moment of inertia, If = I + m * R2
Consider that the merry-go-round rotates with an angular velocity
Final angular momentum, Lf = If *
= (I + m * R2) *

Using the conservation of angular momentum, Li = Lf
m * v * R = (I + m * R2) *
= (m * v * R) / (I + m * R2)